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J O U R N A L O F M A T E R I A L S S C I E N C E 37 (200 2) 1649– 1660

Simulation of porosity by microballoon dispersion

in epoxy and urethane: mechanical

measurements and models

M . A . E L - H AD E K , H . V . T I P P UR∗

Department of Mechanical Engineering, 202 Ross Hall, Auburn University, AL 36849, USAE-mail: [email protected]

Effective mechanical properties of microballoon-dispersed epoxy and urethane are studied

under quasi-static and dynamic loading conditions. Elastic modulus measurements of

these mixtures over a volume fraction range of 0–0.45 are in good agreement with

Hashin-Shtrikman lower-bound predictions for two-phase mixtures comprising of randomlydistributed spherical pores in an elastic matrix. The measurements have also been

predicted accurately by a LEFM based pore-flaw model for a selected flaw size to pore size

ratio. These imply that the microballoons offer negligible reinforcement due to extremelysmall wall thickness to diameter ratio. Accordingly, feasibility of using these materials to

simulate controlled porosity for tensile strength and fracture toughness modeling isexplored. Measured tensile strength and fracture toughness values decrease monotonically

similar to the Young’s modulus variation with volume fraction of microballoons. Guided by

the measurements linear elastic models for porous materials that predict tensile strength

and fracture toughness of these mixtures are proposed and validated. The tensile strength

predictions are in very good agreement with measurements for both epoxy and urethanecompositions. The quasi-static crack initiation toughness prediction captures the

measurement trends rather well in both cases. The agreement between the measurements

and predictions are modest for epoxy matrix while they are good for urethane

compositions. Based on fracture surface micrography, an empirical corrective procedure isadvanced to improve the agreement between the measurements and the model. The

dynamic crack initiation toughness measurements for epoxy, on the other hand, are inexcellent agreement with the predictions. C 2002 Kluwer Academic Publishers

1. IntroductionPorosity in dense engineering materials is often unde-sirable for load-bearing applications. In other instancesporosity is highly beneficial for weight and cost reduc-

tion, enhancing damping and thermal characteristicsand, improving specific strength. Examples of naturalprocesses taking advantage of distributed porosity, asin human bone [1] and wood, are also abundant. In-spired by these biological materials, attempts to create

porous metallic implants of high degree of compatibil-ity with human tissue are also reported [2, 3]. However,research in this area continues as the mechanical prop-erties of these deteriorate to a greater proportion withthe introduction of porosity [4, 5].

Designing for optimum specific stiffness and/or spe-cific strength is an important goal in engineering [6]where modeling mechanical properties of porous ma-terials assumes great deal of significance. Influence of

porosity on brittle systems such as ceramic matrix com-posites is well documented in a recent monograph byRice [7]. Among the early investigations into the me-

∗Author to whom all correspondence should be addressed.

chanical characterization of cellular materials, Gibsonet al. [8] have presented mechanics based analysesof two-dimensional honeycomb structures. They have

considered bending, buckling and plastic collapse of cell walls (‘struts’)for developing expressions for effec-tive mechanical properties. It is shown that ratio of theproperties of the cellular material relative to that of the

‘strut’ are of the form C (ρ/ρs )q , where ρ is the effec-tive density, ρs is thedensity of the wall material, C andq are constants dependent on cell geometry. Using di-mensional arguments, Gibson and Ashby [9] have sub-sequently extended this concept to three-dimensional

cellular materials where the constant C is determinedexperimentally. In their later work, Maiti et al. [10],have introduced a semi-empirical relationship that alsoshows that fracture toughness ratio of the cellular mate-rial relative to its wall properties to depend on the den-

sity ratio. Krstic and Erickson [11, 12] have taken a lin-

ear elastic fracture mechanics approach for predictingYoung’s moduli in elastic porous solids wherein cylin-drical and spherical pores are assumed to possess radial

0022–2461 C 2002 Kluwer Academic Publishers 1649



and annular flaws, respectively. Using elasticity solu-tions for cavities in finite size bodies and crack open-ing displacement concept, models have been developedand verified with experimental data. Effective elasticmoduli of metal-matrix composites with damaged par-

ticles are estimated by Tong and Ravichandran [13]using modified micro-mechanics procedures in uniax-ial and hydrostatic tensile fields which can also be ap-

plied to situations with voids as well as damaged parti-cles. Nielsen [14] has established relationships betweenYoung’s modulus and strength of porous materials foraiding non-destructive evaluation of porous materials.In all these works, material properties have been shownto monotonically decrease with volume fraction of the

pores in the matrix.Development of materials having controlled poros-

ity (pore shape, size and volume fraction) for modelingand simulation (with the exception of honeycomb and

foam materials) purposes is quite challenging. Poly-mer burnout method has been utilized by Zimmermanet al. [15] to produce spherical pores of mean diameter

100µm in alumina. Using these samples they havestud-iedcrackinstability ensuing from pore stressconcentra-

tion near a free edge. Mechanical response of aluminumMMC with microballoon of 1–2 mm diameter and wallthickness to diameter ratios of the order of 0.1 are stud-ied under uniaxial compression by Kiser et al. [16].High degree compressive energy absorption duringfail-

ure has been observed. Parameswaran and Shukla [17]have used relatively thick-walled cenospheres (meandiameter 127 µm and wall thickness 10–15 µm) in apolyester matrix to create functionally graded material.

They have measured effective Young’s moduli, failure

strengths and fracture toughness for different filler vol-ume fractions. Although particulate composite densitydecreaseswith volume fraction, increasing tensile mod-

Figure 1 SEM micrographs of microballoons.

ulus and fracture toughness suggest reinforcement of the matrix by the filler particles in case of Ref. [17].

In the current work, mechanical properties of microballoon-dispersed epoxy and urethane are mea-sured. Apparent tensile modulus, tensile strength and

fracture toughness are measured for different volumefractions in the range 0 to 0.45. Based on the measure-ments feasibility of simulating controlled porosity in

these polymers is examined by evaluating the measure-ments relative to micromechanics models reported inthe literature. Subsequently, simple analytical modelsare proposed for predicting tensile strength and fracturetoughness of porous brittle materials and validated bythe measurements.

2. Material preparationTwo types of polymeric sheets infiltrated uni-formly but randomly with microballoons were pre-

pared. The volume fraction, V f (= V microballoons/

(V matrix

+V microballoons)), of the microballoons in these

sheets ranged between 0 and 0.45. The microballoonsused in this investigation were commercially avail-able hollow soda-lime glass spheres of mean diameter∼60 µm and wall thickness of the order of 100 nm(∼400 nm). A typical micrograph of these fillers isshown in Fig. 1. As it will be demonstrated in thesubse-quent sections, extremely small wall thickness relative

to the diameter these fragile fillers make it suitable formodeling the mixture as a porous material with a rel-atively high degree of accuracy under tensile loadingconditions. Other physical and mechanical propertiesof the constituents are listed in Table I.

Two different thermoset polymers namely, two-partepoxy and urethane were used as matrix materials inthis investigation. Each of these resins have relatively

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TA BL E I Properties of the constituents

Properties Microballoonsa Epoxy Urethane

Mean size ∼60 µm – –

Wall thickness ∼400 nm – –

K Ic (MPa√

m) 0.7 1.1± 0.1b –

Tensile modulus (MPa) – 3016b –

Tensile strength (MPa) – 58c 37d

Density (g/cc) 0.13 1.18 1.08d

Viscosity (centipoises) – 213c

546.7d

a3M Corp., Minneapolis, MN.bReported by R. Young and P. Beaumont [17].cBuhler Inc., USA.dCONAP Inc., USA.

low viscosity at room temperature and long duration(72 hours for epoxy and 168 hours for urethane) cur-ing profiles thereby minimizing residual stresses in the

cured material. Manufacturer recommended ratios of resin and hardener were used for material preparation.Suf ficient care was exercised to prevent trapping airpockets in the mixture. Once the mixture gelled par-tially, it was poured into 100 mm× 76 mm× 6 mmsize

mold and allowed to completely cure. Measurement of longitudinal wave speeds (∝ [

√ ( E /ρ), ν)], E , ν and

ρ being Young’s modulus, Poisson’s ratio and density,respectively) using ultrasonic pulse echo technique atdiscrete locations of each cured sheet was used as a

means of ensuring homogeneity of the material. Sheetswith wave speed variation of ±2.5% over the length of the sheet were considered homogeneous and utilizedfor further investigation.

3. Material characterization3.1. Tensile testsThe stress-strain responses of materials with different

volume fractions were measured from uniaxial tensiletests (Fig. 2) performed accordingly to ASTM standardD-638 for rigid plastics. Standard dog-bone sampleswere machined according to the dimensions shown inFig. 3. The tests were carried out at room temperature

using Instron Testing Machine (Model 4465) fitted witha 5 kN load cell operating in the displacement controlmode. The cross-head speed was 0.25 mm/min. The

tests were repeated with three samples for each vol-ume fraction. The stress-strain responses for un-filledepoxy and urethane are shown in Fig. 4. Urethane issignificantly compliant compared to epoxy and has arelatively nonlinear elastic response. For selected mix-

Figure 2 Tension test specimens.

Figure 3 Uniaxial stress-strain response for unfilled epoxy, urethane.

tures of epoxy and urethane with microballoon volumefraction, stress-strain responses are shown in Fig. 4aand b. The plots are essentially linear for epoxy mix-tures while modest nonlinearity prior to failure is evi-

dent in urethane compositions.As the volume fraction of the microballoons in

the matrix increase, the failure stresses and strainsdecrease. Further, this leads to the reduction of the

Young’s modulus and an increase of the elastic strainat a given stress. This has been explained by Krsticand Erickson [12] as a consequence of crack opening

displacement caused by the presence of radial and/orannular cracksassociated with pores. Thefailure strains

are noticeably higher in urethane mixtures compared toepoxy samples. The opposite is true when the tensilestrengths of these two materials are considered.

3.2. Elastic modulus—quasi-staticmeasurements

Young’s moduli were determined from uniaxial stress-

strain responses and are plotted as solid symbols inFig. 5a and b for epoxy and urethane samples. Thevalues of the Young’s moduli for unfilled polymers inboth cases agree well with the values reported in the lit-erature [18]. The values decrease monotonically with

microballoon volume fraction over the entire range. Ap-proximately 60% reduction in the Young’s modulus be-tween infiltrated (V f ∼ 0.45) and unfilled polymers areevident.

To investigate the feasibility of using microballoon

dispersion as a means of introducing controlled poros-ity in these polymers, comparison of the measure-mentswith micro-mechanics predictions based Hashin-Shtrikman lower bound estimation [19–21] was carried

out. It should be noted that, for spherical filler particles

in two-phase compositions, lower bounds are known tooffer accurate prediction of apparent Young’s modulus[20] and hence recommended. For a two-phase mixturecomprising of matrix and spherical fillers, the apparent

bulk ( B) and shear (µ) moduli are expressed in terms

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Figure 4 Stress-strain curves for microballoon-dispersed (a) epoxy compositions, (b) urethane compositions.

of the corresponding properties of the matrix (subscript‘m’) and filler (subscript ‘ f ’) as,

Bc = B f +V f

1

Bm − B f

+ 3(1− V f )

3 Bm + 4 B f

, (1)

µc = µ f +V f

1

µm − µ f

+ 3(1− V f )( Bm − 2 µm )

5 µm (3 Bm + 4 µm )

,

(2)

where V f is the filler volume fraction and subscript ‘c’denotes the properties of the mixture. Now, bulk and

shear modulus estimates of the porous material in termsof pore volume fraction and properties of the matrixwere determined by setting values of B f and µ f equalzero in Equations 1 and 2. Subsequently, the Young’s

Figure 5 Comparison between predicted and measured quasi-static Young’s moduli for microballoon-dispersed (a) epoxy, (b) urethane compositions.

modulus was determined from the apparent bulk andshear moduli using,

E c =9 Bcµc

3 Bc + µc

. (3)

The comparison between micro-mechanics predic-tions (solid line) and experimental measurements (solidsymbols) are shown in Fig. 5. Excellent agreement be-tween the two sets of data is evident over the entire

range of volume fractions studied. The maximum dif-ference between the predictions and measurements ineach case is less than 10%. The agreement also suggests

negligible reinforcement of both epoxy and urethanecompositions by the microballoons and hence allows

further investigation of these compositions as porousmaterials.

Also shown in Fig. 5 are the Young’s modulus pre-dictions by Gibson and Ashby [8] for cellular solidsandmodified Mori-Tanakaapproach [22] for two-phase

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mixtures with filler phase properties set equal to zero.Substantial under prediction of Young’s moduli by thelatter is evident for both epoxy and urethane mixtures.On the contrary the cellular material model [8] overpredicts the measurements consistently but fairly ac-

curately. As discussed in the Introduction, a pore-flawmodelthat accounts for stress concentration effects neara pore along with crack opening displacements has

been proposed by Krstic and coworkers [12, 23, 24]to explain the Young’s modulus reduction. In these, theYoung’s modulus of the porous material is given by,

E c = E m

1+ 16 N p R3

1 − ν2

m

3

φ

−1

, (4)

where φ is a function that depends on the flaw size s

and the pore radius R as follows:

φ

= 1

+

s

R

3

+

9

2(7− 5ν)

1+ s R

2

+

4 − 5ν

2(7− 5ν)

,

and

N p =3V f

4π R3. (5)

In Equations 4 and 5, the flaw of size s is assumed tobe a hemispherical annular crack subjected to the stressfield in the vicinity of a spherical pore in an elasticbody. The Young’s modulus values calculated using

this model were found to agree well with the measuredones when s/ R∼ 0.26 and s/ R∼ 0.22 for epoxy andurethane compositions, respectively. The comparison isshown as broken line in Fig. 5. It should be pointed out,however, that in the absence of systematic guidelines on

the choice of s/ R values and the experimental dif ficulty

Figure 6 Comparison between predicted and measured dynamic Young’s moduli for microballoon-dispersed (a) epoxy, (b) urethane compositions.

in determining such as value, the values selected werebased on trial and error.

3.3. Elastic modulus—dynamicmeasurements

The values of apparent dynamic Young’s moduli ( E d )

for epoxy and urethane compositions were also deter-mined for completeness. These values were determinedby measuring longitudinal (C l ) and shear (C s ) wavespeeds,

C 2l = E d

ρ

(1− ν)

(1+ ν)(1− 2ν), C 2s =

E d

ρ

1

2(1+ ν),

(6)

in these materials using ultrasonic pulse-echo tech-nique. In the above,ρ and ν denote apparent density and

Poisson’s ratio, respectively. The ultrasonic transducersof crystal diameter 3 mm and 5 mm, operating at fre-quencies of 10 MHz and 2.25 MHz, respectively, wereused for epoxy and urethane. The measurements wereused in conjunction with measured densities of differ-

ent mixtures and equations (6) to determine dynamicYoung’s modulus and Poisson’s ratio. Incidentally, thevariation in the value of Poisson’s ratio was found to besmall in the range 0.35± 0.01.

The variation of apparent dynamic Young’s moduli

with volume fraction is shown in Fig. 6 and is similar tothe ones seen for quasi-static counterparts. Due to therate dependency of the matrix materials, the values of Young’s modulus are consistently higher for both types

of mixtures when compared to their quasi-static coun-terparts shown in Fig. 5. The reduction in the values

of E d between unfilled and the mixture with V f = 0.45for both epoxy and urethane is again ∼60%. Shown inFig. 6, are the Hashin-Shtrikman predictions [20] for

two-phase mixtures with the elastic properties of thefiller set equal to zero, thereby representing predictionsfor porous materials. Excellent agreement between the

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Figure 7 The variation of tensile strength of microballoon-dispersed (a) epoxy, (b) urethane compositions with volume fraction.

measurements and predictions are seen and reinforce-ment of the matrix by the filler, if any, is therefore neg-ligible. Also, shown in Fig. 6 are the predictions basedon the pore-flaw model (Equation 4) for the same s/ R

values used in quasi-static counterparts. Evidently, the

agreement between the two is good.

4. Tensile strengthFrom the stress-strain responses shown in Fig. 4, ap-

parent tensile strengths, (σ c)cr, of epoxy and urethanecompositions were also measured. The variation of ten-sile strengths with volume fraction of microballoons isshown in Fig. 7 by solid symbols. Nonlinear mono-tonic reduction in the apparent tensile strength withfiller volume fraction over the entire range is observed.

Maximum strength reduction of approximately 60% inboth epoxy and urethane relative to unfilled matrix areevident. Notably, the trends are similar to the variationof Young’s modulus shown in Fig. 5. This phenomeno-

logical observation forms the basis for a simplified pre-dictive model for tensile strength described next.

Predictive models for the apparent tensile strength of particle reinforced composites are very limited (Nielsen[14], Gibson and Ashby [9]). In the present work, a

strain energy based model is proposed for predictingthe strength of brittle porous bodies. Consider a porouslinear elastic sheet, shown schematically in Fig. 8. Letthe material have a random distribution of sphericalcavities throughout the matrix. Now, consider a tensile

stress σ c acting on this sheet normal to the nominalcross-section area A(=b× t , width and thickness, re-spectively). The corresponding apparent tensile strainbe εc. Within the control volume ( L × b× t ), the aver-age reduced cross-section area normal to the far-field

stress be A(=b× t ) due to the presence of cav-ities in the material with • denoting ensembled av-erage. Then, recognizing that the overall strain energystored in the porous composition is equal to that in the

matrix material, for linear elastic material behavior and

plane strain conditions,

AL

1

2σ cεc

= A L

1

2σ xε x +

1

2σ yε y +

1

2σ x yγ x y

m

(7)

where x , y denote in-plane Cartesian coordinates of theporous sheet as shown. By recognizing that strain en-

ergy in the matrix can be approximated by the dominantfirst term to a high degree of accuracy (see, Appendix),the right hand side of Equation 7 can be simplified as,

AL

1

2σ cεc

≈ A L

1

2σ xε x

m

≡ A L

1

2σ mεm

.

(8)

In the above, for simplicity the subscript ‘ x’ has beendropped and σ m and εm denote the average valuesof matrix stress and matrix strain components, respec-tively. The above simplification implicitly takes into

account the reduction in stress concentration effects

with decreasing separationdistance between pores [25].Now, recognizing that σ c A=σ m A, Equation 7 be-comes,

εc ≈ εm. (9)

Equation 9 implies that the apparent strain in the porous

medium is approximately equal to the average ma-trix strain component in the x-direction. Accordingly,Equation 7 can be expressed as,

σ c=

E c

E m σ m ⇒

(σ c)cr

= E c

E m(σ m)cr, (10)

where (σ c)cr denotes the apparent tensile strength of themixture and (σ m)cr is the tensile strength of thematrix.

Thus, the tensile strength of the porous composition is

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Figure 8 Uniaxial tension model to predict tensile strength.

described in terms of the Young’s modulus ratio of theporous and matrix materials and, tensile strength of the matrix. It should be noted that in Equation 10, thematrix strength of the matrix material being constant,

the strength of the mixture is dependent on the volumefraction of the cavities in the mixture through Young’smodulus of the mixture ( E c ≡ E c(V f )). This is similarto the observations made by Nielsen [14] that Young’s

modulus and strength of porous materials are directlyrelated.

4.1. Experimental validationThe validity of the proposed model is demonstrated inFig. 7 forboth the polymers. Evidently, good agreement

of generally better than 10% deviation between themeasurements and Equation 10 (solid line) is seen forthe entire range of volume fractions studied. Accord-ingly, one could conclude, that despite its simplicity,the proposed model is indeed able to capture the essen-

tial features of tensile strength behavior effectively forthese two material systems modeled as porous solids.

Krstic [24] has also proposed an expression for pre-dicting the tensile strength of porous materials with

spherical cavities of diameter D as,

(σ c)cr =1

φ

π E cγ

D

1+ s

R

1− ν2

c

1/2

(11)

where φ is geometric function given by Equation 5,and γ is the fracture energy of the matrix. Again, lin-ear elastic fracture mechanics approach was used inderiving Equation 11 where the existence of annu-

lar flaw (of size s) around the spherical pore was as-sumed. The predicted values of the tensile strength us-ing s/ R∼ 0.26 and s/ R∼ 0.22 for epoxy and urethanecompositions, respectively, however, deviate substan-

tially from the measurements and therefore not shown.

Further, predictions based on Equation 11 for the s/ R

values mentioned in the previous section over estimatethe strength. In order for the model to agree with themeasurements, one has to select a significantly differ-

ent value of s/ R(∼0.7) from the one used for predictingYoung’s modulus in Figs 5 and 6. One could, however,see the applicability of pore-flaw model (Equation 4)for predicting strength in from a different perspective.

Since the predicted values of E c of the porous mate-rial based on Equation 4 agree well with the measure-ments, one could utilize Young’s modulus predictionsin Equation 10 for successfully predicting the measuredstrength (and Equation 15 for predicting critical stress

intensity factors, to be discussed in the next section).This in turn would incorporate the pore-flaw concept inthe strength prediction process.

5. Critical stress intensity factorsAnother important aspect of the mechanical behaviorof porous materials is the fracture response. The crack

initiation toughness or critical stress intensity factors(K I)cr in microballoon filled epoxy and urethane withrespect to the pore volume fraction were determined

using three-point bending tests. Edge notched beams,shown schematically in Fig. 9, were prepared for se-lected volume fractions up to 0.45. In each specimen, an

Figure 9 Loading configuration and geometry of single edge notched

specimen.

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initial notch of root radius 75 µm and length 5 mm wasintroduced using a high-speed diamond-impregnatedcircular saw. The notch was subsequently extended intoa sharp crack by gently tapping-in a wedge into themouth of the notch forcing the crack tip to ‘pop’ by a

few millimeters and arrest. Care was exercised to en-sure the tip of the wedge from contacting the tip of thenotch during crack extension, thereby preventing un-

due residual deformations at the crack tip. The crackedsamples were loaded in an Instron Testing machine withdisplacement controlled loading and quasi-static load-ing history was recorded using a PC-based DAQ sys-tem. Typically five samples were tested foreach volumefraction and the failure load ( Pcr) in each case was used

to determine crack initiation toughness using [26],

(K I)cr = f

a

W

Pcr S

BW 3/2, (12)

with the geometric factor f (a/W ) given by,

f

a

W

=

3

a

W

1.99 − a

W

1 − a

W

− 2.15 − 3.93

a

W + 2.7

a

W

2

2

1 + a

W

1 − a

W

3/2.

In the above, a, S, B and W are crack length, span,thickness and height of the beam samples (see Fig. 9),respectively. The apparent critical stress intensity fac-

tors, [(K I)cr]c, determined from the above are plottedas a function of volume fraction of the microballoonsin Fig. 10a and b for epoxy and urethane mixtures,respectively. The measurements monotonically de-crease with increasing volume fraction of the mi-

croballoons. This trend is similar to the observation of other investigators for brittle ceramics having different

Figure 10 Predicted and measured effective crack initiation toughness of microballoon-dispersed (a) epoxy, (b) urethane.

degrees of porosity [27, 28]. The values of [( K I)m ]cr

for unfilled epoxy and urethane are approximately 1.2MPa

√ mand1.45MPa

√ m, respectively. The measured

value for the epoxy is also in good agreement with thosereported in the literature for quasi-static loading con-

ditions [18]. For epoxy, the reduction [( K I)cr]c for V f

ranging 0–0.45, is nearly 40% while for urethane it isabout 58%. These percentage reductions, particularly

in case of epoxy, are somewhat lower than the ones seenfor tensile strength over the same V f range. Crack ini-tiation toughness being a local material characteristic,fracture surface micrographs were studied in order toreconcile this observation and will be discussed later.

As done with tensile strength, a simple model to pre-

dict critical stress intensity factors for brittle porousmaterials based on linear elastic fracture mechanicsis proposed. The phenomenological observation that[(K I)cr]c values decrease monotonically with microbal-loon population similar to Young’s modulus and tensile

strength values is utilized in proposing this model. The

apparent stress intensity factor for a generic mode-Icrack in a microballoon-dispersed mixture is,

(K I)c = σ c√ πa, (13)

where a is the flaw size. Now, from Equation 10 σ c canbe expressed in terms of σ m to get,

(K I)c = E c

E mσ m

√ πa. (14)

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The critical value of apparent stress intensity factor forthe porous composition [(K I)cr]c can then be expressedas,

(K I)cr

c= E c

E m(σ m

√ πa)cr =

E c

E m

(K I)cr

m, (15)

where [(K I)cr]m is the crack initiation toughness of

the matrix material. Thus, Equation 15 suggests that[(K I)cr]c variation would be similar to that of theYoung’s modulus of the porous solid since Young’s

modulus and crack initiation toughness for the matrixare constant. This is consistent with the earlier observa-tion that experimentally measured values for the mix-tures show monotonic reduction in [(K I)cr]c similar to

the trends for E c in both epoxy and urethane.

5.1. Experimental validation—quasi-staticloading

Solid lines in plots in Fig. 10a and b were obtained fromEquation 15 and are superimposed on the experimen-

tal measurements. In each case Equation 15 providesa conservative estimate of the apparent crack initiationtoughness value for all volume fractions considered.Evidently, the agreement between the measurements

and the predictions is only modest for epoxy with bet-ter agreement seen at lower volume fractions and in-creasing deviation with V f . In case of urethane, on theother hand, the agreement is rather good over the entirerange.

As alluded to earlier, fracture behavior being a localcrack tip response, micrographs of the fracture sur-

faces were examined to further ascertain the observa-tions. The fractured surfaces of microballoon-dispersedepoxy and urethane with a volume fraction of 0.25

were studied using SEM. Typical micrographs for thesame are shown in Fig 11a and b. The fracture sur-face morphology reveals that the microballoons havecompletely fractured in both types of matrix. Further,

cleavage fracture in epoxy matrix is readily evident

Figure 11 SEM micrographs for (a) epoxy, (b) urethane, infiltrated with microballoons at 0.25 volume fraction (quasi-statically fractured surfaces).

while in case of urethane it is relatively fibrous dueto higher degree of material nonlinearity. Interestingly,in case of epoxy mixture, in addition to hemispher-ical microballoon fracture, some also appear to haveseparated from the surrounding matrix material. This

suggests additional dissipation of energy (in additionto matrix and microballoon fracture) during crack ini-tiation and growth. On the contrary, microballoon in-

terfacial separation is minimal in case of urethane mix-ture. Since all other parameters such as microballoonsize, material processing and experimental methods areidentical, interfacial energy dissipation is potentially atthe root of somewhat larger deviation between exper-iments and predictions in Fig. 10a. In this context, it

should be noted that since nominal fracture toughnessvalue of soda-lime glass (∼0.7 MPa

√ m [29]) is signif-

icantly lower than that for both the matrix materials andthe wall thickness of microballoons being of the order

of 100 nanometers, the energy dissipated through mi-croballoon fracture would be a negligible fraction of thetotal energy for fracture. Accordingly, interfacial sepa-

ration is a likely source of deviation between measuredand predicted values of [(K I)cr]c. An empirical way of

accounting for the deviation is to modify Equation 15as,

(K I)c

cr− K cor ≡

(K I)c

cr= E c

E m

(K I)m

cr, (16)

where K cor denotes a stress intensity factor correctionthat takes into account interfacial separation seen insome microballoons. Thus, [(K I)c]cr in Equation16 de-note approximate values of apparent stress intensity

factors at crack initiation in the absence of any micro-interfacial separations. Now, assuming K cor to be of the form, K cor

∼=αV f |K int|, where α denotes a knownfraction of microballoons (0≤α≤ 1) which show in-terfacial separation along a typical crack front and|K int| represents an ‘average’ value of interfacial crack initiation toughness. Here α can be determined fromthe micrographs by the percentage of total number of

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particles showing evidence of separation from the ma-trix. To proceed further, a value of |K int| is necessary.It is widely recognized that interfacial fracture processis inherently mixed-mode in nature [30] and interfa-cial crack initiation toughness generally depends on the

local mode-mixity [31]. However, for simplicity, |K int|was approximated to be constant and equal to the frac-ture toughness of glass. From the micrographs it was

determined that α is ∼0.4 in case of epoxy and ∼0.2 incase of urethane. Taking these factors into account, val-ues of K cor(V f ) and corresponding values of [( K I)c]cr

were evaluated. The values of the critical stress inten-sity factor based on Equation 16 are shown in Fig. 10 asopen symbols. The agreement between the open sym-

Figure 12 Apparent dynamic crack initiation toughness of

microballoon-dispersed epoxy.

Figure 13 SEM micrographs for epoxy infiltrated with microballoons at 0.25 volume fraction. (dynamically fractured surface).

bols and the predictions (solid line) are substantiallyimproved when compared to the solid symbols. A moresophisticated analysis could potentially mitigate the de-viation to a greater extent. Alternatively, improving theinterfacial strength between the microballoon and the

epoxy matrix could eliminate the problem as well.

5.2. Experimental validation—dynamic

loadingDynamic values of critical stress intensity factors

([(K d I )c]cr) for microballoon-dispersed epoxy were also

evaluated. Homogeneous beam (150 mm× 20 mm×6 mm) samples (four specimens for each volume frac-tion) with single edge notches were prepared. The

notches were extended into sharp cracks as describedearlier. The samples were subsequently impact loaded(impact velocity of 1.5 m/s) in three-point bending con-figuration in Instron-Dynatup - 8210 instrumented droptower. This was achieved by releasing a 90 N (20 lb)

deadweight along guide rails from a designated height.Integral to the deadweight is a cylindrical tup, with a

hemispherical end of radius 12.7 mm, that impacts thespecimen directly on the edge opposite to the crack. Af-fixed to the tup are sensors that feed information such

as time, load and displacement history experienced bythe tup to a PC-DAQ system. The maximum load reg-istered at the tup was used to assess the dynamic crack initiation toughness using Equation 12. (The rate of

increase in stress intensity factor prior to initiation inall specimens tested was 19± 1.8 MPa-

√ m/ms). The

measured values of [(K d I )c]cr for different volume frac-

tions of microballoons are plotted as solid symbols inFig. 12. Evidently [(K d

I )c]cr-V f variations are quali-

tatively similar to the ones obtained for quasi-staticloading (Fig. 10). That is, values of [(K d

I )c]cr monotoni-cally decrease with V f and an overall reduction of about

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60% between unfilled and the mixture with V f = 0.45is seen. The percentage reduction for dynamic fracturetests, however, is comparable to the ones for Young’smodulus (Fig. 5a) and tensile strength (Fig. 6a). This,however, is unlike the quasi-static results shown in

Fig. 10a. It should be noted that values of [(K d I )c]cr

for each V f is significantly higher than the correspond-ing value [(K I)c]cr obtained under quasi-static loading

conditions, due to the loading rate dependent behaviorof epoxy.Also shown in Fig. 12 are the predicted values (solid

line) of [(K d I )c]cr from Equation 12 for different vol-

ume fraction of microballoons. In these calculations,the dynamic Young’s modulus values from Fig. 6a

were utilized. Evidently, the agreement between themeasurements and the predictions is very good overthe entire range of volume fractions studied. This isunlike the values for quasi-statically fractured beams(Fig. 10a). To further reconcile this observation, a dy-

namically fracturedsample surface with a microballoonV f

=0.25 was examined microscopically. The corre-

sponding micrograph is shown in Fig. 13. Interestingly,unlike in Fig. 11a, the interfacial separation of the mi-

croballoons seen under quasi-static loading conditionsis non-existent in the dynamic case and hence the bet-ter agreementbetween the model andthe measurementsunder dynamic conditions. Otherwise, the fracture sur-face is relatively more rugged when compared to quasi-

statically fractured surface accounting for higher ex-penditure of energy during fracture.

6. Concluding remarksPossibility of dispersing microballoons in epoxy and

urethane to simulate controlled porosity in polymers ina simple and cost-effective way is examined. To thisend, homogenous samples with different microballoonvolumefractions, ranging from 0 to 0.45, were preparedand tested. Uniaxial tension tests on dog-bone samples

and three-point bending on SEN samples were carriedout to arrive at the following conclusions:

• The Young’s modulus was found to decreasemonotonically over the entire range of microbal-loon volume fractions. The measurements, quasi-static as well as dynamic, are in very goodagreement with Hashin-Shtrikman lower-bound

predictions for two-phase mixtures comprisingof matrix and randomly distributed sphericalvoids/pores. The pore-flaw model proposed byKrstic and Erickson show good agreement with

the measurements as well for flaw size to pore sizeratio of 0.26 and 0.22 for epoxy and urethane re-spectively.

• The tensile strength was found to decreases mono-tonically with increasing microballoon volume

fraction and the trends are akin to the ones ob-

served for Young’s modulus. Guided by the exper-imental observation,a linearelastic model based onstrain energy balance has been proposed for pre-dicting tensile strength of brittle porous materials.

The model generally predicts the measurements to

within 10% for both epoxy and urethane composi-tions, over the entire range of volume fractions.

• The critical stress intensity factor values measuredunder quasi-static loading conditions were found

to continuously decrease with microballoon vol-ume fraction as in case of tensile modulus andstrength. Again, based on experimental evidence,a predictive model using linear elastic fracture me-

chanics has been introduced. A somewhat large de-viation between the measurements and predictionsin case of epoxy is explained by micrographs thatreveal additional expenditure of energy by interfa-cial separation between microballoons and the ma-

trix. Otherwise, the microballoons generally showhemispherical fracture in both matrix materials.Anempirical correction factor based on this observa-tion has been introduced to reduce the differencebetween the predictions and measurements.

• The critical stress intensity factor values measuredunder impact loading conditions also monotoni-cally decrease with microballoon volume fraction.

Unlike quasi-static results, microscopic examina-tion of dynamically fractured surfaces generallyshowed hemispherical fracture of microballoonsand interfacial separation was non-existent. Ac-cordingly, the proposed predictive model for brittleporous materials for critical values of apparent

stress intensity factors agrees well with the dy-namic measurements.

• The proposed models for tensile strength and frac-ture toughness of mixtures suggest the variation tobe directly proportional to their Young’s moduli

with volume fraction of the pores. The measure-

ments using in microballoon-dispersed epoxy andurethane support the model.

AppendixThe approximation that under uniaxial remote loading,the energy density U (= 1

2(σ xε x + σ yε y + σ x yγ x y )) is

given by the dominant term U (=12σ xε x ) was validated

through elasto-static finite element modeling of aporous sheet with an array of distributed cylindrical

Figure A1 The dimensionless finite element model, where the element

size is 40 µm.

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Figure A2 The ratio U /U with respect to volume fraction.

cavities. The finite element model used for this ver-ification is shown in Fig. A-1. Due to symmetry, afourth of the model was used in the computations. Two-dimensional plane strain simulations with cylindricalcavities were carried out instead of 3-D simulations in-

volving spherical cavities for (a) simplicity and (b) amore conservative estimate of the error involved in theapproximation. (Here it should be noted that analyticalsolutions for the limiting case of single cavity in an infi-nite sheet subjected to uniaxial tension indeed suggest a

relatively benign stress concentration effect for spher-ical cavity (K t = 2) compared to a cylindrical cavity(K t

=3).)Accordingly, totalstrain energy density in the

matrix U and its approximated value U were evaluated

for different volume fractions of porosity in the controlvolume (shown by the broken line). Fig. A-2 showsthat, the ratio U /U is in the range of 0.9–0.92 for thevolume fractions studied and hence the assumption isreasonable.

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Received 20 December 2000

and accepted 22 October 2001

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